Following the seminal papers of Garriga, Vilenkin and collaborators [1]}, [2]}, cosmological evolution on the landscape is described by a
Markov process. Because of terminals, this is technically an absorbing Markov process — detailed balance is explicitly violated, hence the dynamics are out-of-equilibrium.
Along a given world-line, the probability \(f_I(\tau _I)\)  to occupy vacuum \(I\)  as a function of the local proper time \(\tau _I\) 
satisfies the master equation
\(\Delta f_I = \sum _J \left( \kappa ^{\text{proper}}_{IJ} - \delta _{IJ} \sum _K \kappa ^{\text{proper}}_{KJ} \right) \Delta \tau _J\, f_J \,,\) 
